We present the first polynomial-time approximation schemes
(PTASes) for the following subset-connectivity problems in
edge-weighted graphs of bounded-genus: Steiner tree,
low-connectivity survivable-network design, and subset TSP. The
schemes run in O(n log n) time
for graphs embedded on both
orientable and nonorientable surfaces. This work generalizes the
PTAS framework from planar graphs to bounded-genus graphs: any
problem that is shown to be approximable by the planar PTAS
framework of Borradaile, Klein, and Mathieu (2007) will also be
approximable in bounded-genus graphs by our extension.